Estimates of \(\Delta _M\)  as graph Laplacians \(L_h\)  defined in terms of \(X_1,X_2,\ldots \)  are extensively used in machine learning and computer graphics to get at the geometry of \(M\)  (cf. [1]}, [2]}.)
For example, \(X_1,X_2,\ldots \)  can be usefully mapped onto a lower dimensional Euclidean space spanned by the first few eigenvectors of \(L_h\)  [1]}.
The consistency of \(\hat{\Delta }_{h}\)  has been extensively studied in the literature [4]}, [5]}, [6]}.
However, these estimates \(L_{h}\)  have not been used to obtain estimates of intrinsic distances in \(M\) .
For example, distances obtained by the previously mentioned dimensionality reduction (cf. [1]}, [8]}.) are Euclidean distances and so cannot possibly estimate manifold distances.
This may be because while nearly every existing application of \(L_h\)  is based on the properties of \(L_h\)  as a linear operator, it is impossible to get the complete geometry of \(M\)  from \(\Delta _M\)  as a linear operator alone.
